Weinstein Applies Physics Gauge Theory to Fix Consumer Price Index

温斯坦应用物理学规范场论修正消费者价格指数 在弯曲纤维丛上对消费者偏好和双边贸易建模，将路径依赖的完整群视为关键经济数据而非分析误差，从而消除了传统价格指数中的巨大失真。 温斯坦提出了一种经济学的规范场论表述，旨在解决经典指数问题，并在不依赖任意基数效用度量的情况下解释消费者偏好的变化。通过将价格-数量向量丛分解为正交的收入子丛和替代子丛，他证明了在规范协变性下，经典价格指数自然会收敛到路径依赖的迪维西亚指数。他还将该模型推广到适用于双边国际贸易的非阿贝尔规范场论，以及由圆周微分同胚群参数化的用于消费者福利的无穷维主丛。最终，他认为现代微分几何的数学工具并非理论物理学所独有，并敦促物理学家摆脱封闭的学术文化，将其方法应用于经济现象。 埃里克·温斯坦提出了经济学的规范场论基础，将1975年的吴-杨字典扩展为连接粒子物理学、微分几何与市场动力学的三向罗塞塔石碑。 (0:00) 包括拉斯佩尔和帕舍在内的经典价格指数公式，一旦在经济状态空间中被表述为规范协变，便会自然坍缩为路径依赖的迪维西亚指数。 (40:40) 消费者价格指数（CPI）向上的替代偏误高估了生活成本，其财政规模堪比社会保障、医疗保健和国防等主要联邦支出项目。 (24:39) 通过将序数偏好叶状结构与价格共变向量空间相交叉，温斯坦构建了一个具有非零曲率的正规联络，该联络解决了偏好变动问题，并将经济周期重构为完整群而非度量误差。 (51:28)

Eric Weinstein argues that economic market dynamics—specifically inflation measurement and shifting consumer preferences—are naturally governed by the mathematical machinery of gauge field theory, fiber bundles, and differential geometry. By extending the 1975 Wu-Yang dictionary beyond physics into economics, he contends that longstanding measurement paradoxes can be resolved while challenging the assumption that modern geometry belongs uniquely to fundamental natural laws. The Index Number Problem and Measurement Bias (5:28) Tracking price levels is a critical practical problem, yet economic measurement is hindered by the classical index number problem, wherein standard metrics yield conflicting results. Ragnar Frisch proved in 1930 that no price index can satisfy even three of Irving Fisher's intuitive baseline axioms, such as time reversal and circularity. Weinstein highlights that the Laspeyres index—historically used for the United States Consumer Price Index (CPI)—evaluates price changes against a fixed base-period basket, failing to account for consumer substitution toward relatively cheaper alternatives when relative prices change. Constructing an Economic Connection on Fiber Bundles (30:47) Weinstein diagnoses the flaw in standard indices like Laspeyres as evaluating vector quantities across different points in time using an unexamined, trivial connection. Over a base space of price and quantity vectors minus points where expenditure is zero, Weinstein defines an economic connection that projects the trivial derivative onto four canonically invariant sub-bundles. These subspaces correspond directly to the classic microeconomic distinction between income effects and substitution effects. Holonomy and the Rehabilitation of the Divisia Index (40:40) Under this geometric formulation, classical competing indices collapse into the continuous, path-dependent Divisia index once made gauge-covariant. Although mainstream economists historically rejected the Divisia index because its path dependence creates cycling—where an index does not return to unity even if prices and quantities return to their starting values—Weinstein reinterprets this path dependence as geometric holonomy. Rather than an analytical defect, holonomy captures intermediate market reality and exposure. Non-Abelian Trade and Infinite-Dimensional Preference Bundles (41:58) The framework expands beyond single-agent abelian settings into non-abelian gauge theory. In bilateral international trade between two nations where purchasing power parity fails, parallel transport requires a path-ordered exponential of matrices, producing a 2x2 matrix-valued index in GL(2, R) analogous to non-abelian gauge theories. To resolve the 'changing preference problem,' Weinstein models cardinal utility functions as a principal bundle over the base space of Pareto ordinal preference foliations, with the infinite-dimensional group of orientation-preserving diffeomorphisms acting as the structure group. Philosophical Implications and Academic Insularity (51:28) Weinstein argues that the presence of market prices provides the necessary data to form a canonical, natural connection that resolves changing consumer tastes without relying on unmeasurable cardinal utility labels. He challenges theoretical physicists to reconsider the insularity of their discipline, arguing that their 'monastic culture' has treated gauge theory as a miraculous property unique to fundamental particle physics, rather than recognizing it as a universal geometric toolkit applicable to human-made economic systems. Weinstein acknowledges that practical challenges remain, such as accommodating entirely new products through hedonic characteristic projections or shadow pricing. He leaves theoretical physicists with an open epistemological dilemma: if the geometric framework of the Standard Model and General Relativity applies just as naturally to the messy, human-designed world of economics, physics must reconsider whether these geometric structures are truly fundamental or merely emergent effective tools.

数学物理学者 Eric Weinstein 提出，应将微分几何与规范场论扩展至经济学领域，通过扩充 1975 年物理学与几何学间的“吴-杨字典”，建立包含纤维丛、联络与完整群的经济规范理论，以解决指数编制与动态偏好变化等经典难题。 指数编制困境与传统度量的系统性偏差 (5:28) 在宏观经济与金融市场中，衡量物价通胀面临经典的“指数编制问题”（Index Number Problem）。传统测度如拉斯佩尔指数（Laspeyres Index）使用基期消费篮子，而帕舍指数（Paasche Index）使用现期消费篮子。早在 1930 年，经济学家 Ragnar Frisch 即证明了不可能定理：欧文·费雪提出的看似合理的公理体系，在数学上无法同时满足其中的任意三条。更严重的是，美国消费者价格指数（CPI）等传统统计指标因忽视消费者在相对价格变化下的替代效应，导致生活成本被长期系统性高估。Weinstein 指出，由于美国税收等级与社保等福利支出直接与 CPI 挂钩，该偏差若换算为联邦财政支出，其规模仅次于社保、医疗与国防，成为变相的“第四大支出项目”，造成数千亿美元规模的财富再分配扭曲。 新古典经济学的防御机制与偏好时变难题 (26:18) 微观经济学以无差异曲面和预算约束的切点来界定消费者的最优化行为，但在处理时间维度上的偏好漂移时陷入了理论死角。以 Gary Becker 和芝加哥学派为代表的新古典传统，为了防止无差异曲面在时间维度相交打破传递性公理，选择将“偏好稳定不变”强行假定为经济学定义的一部分。Daniel McFadden 等学者亦曾指出，经济学界因数千年来无法定量处理偏好变化，倾向于回避该现实。当面对健康意识提升导致牛肉与鸡肉消费转移等实际问题时，传统模型只能将其归咎于相对价格波动。Weinstein 认为，真正解决偏好变化问题，必须在不依赖具有基数性质的“效用单位”（utils）的前提下，建立一套坐标无关的内蕴几何映射，实现不同时间切片上序数偏好叶状结构之间的严格对应。 理论一：有限维阿贝尔规范理论与迪维西亚指数 (30:47) 针对跨期乘积中时间不同步的张量积缺陷（如以现期价格乘以基期消费量），Weinstein 构建了一个平凡向量丛，其底空间为商品数量空间与价格对偶空间的乘积并剔除奇异集。在每一个底空间点上，纤维被典则地正交分解为对应“收入效应”的平行子空间和对应“替代效应”的对偶子空间。通过对平凡外微分投影，他定义了“经济联络”（economic connection）以规范平行移动。在此框架下，传统的拉斯佩尔、帕舍及托恩奎斯特指数在规范协变要求下均收敛为 1925 年提出的迪维西亚指数（Divisia Index）。经济学界曾将迪维西亚指数的路径依赖性视为分析缺陷，但 Weinstein 阐明其非零曲率正对应着微分几何中的完整群（holonomy），恰恰包含了沿路径累积的真实经济信息。 理论二与三：非阿贝尔贸易模型与无限维主丛 (41:58) 该框架进一步推广至复杂的非阿贝尔与无限维情形。在双边国际贸易场景中，两国价格与消费向量构成 $2 \times n$ 和 $n \times 2$ 矩阵，此时的平行移动演化为不可交换的路径有序指数（威尔逊圈），其完整群属于 $\mathrm{GL}(2, \mathbb{R})$ 矩阵群，成为一种非阿贝尔规范理论。在消费者效用层面，Weinstein 将序数偏好映射空间设为底空间，以保持实直线方向的保向微分同胚群 $\mathrm{Diff}_+(\mathbb{R})$ 为结构群，构造出无限维主丛。通过将效用函数空间与市场可观测的价格空间作笛卡尔积，系统获得了典则存在的内蕴联络，通过戴森级数形式的微扰调整因子，实现了完全独立于人为基数设定的规范不变效用平行移动。 几何完整群的现实意义与物理学的修道院文化 (54:00) 为阐明完整群在现实中的作用，Weinstein 提出了一个三位对冲基金经理在物价量谐振循环周期中的思想实验：在起始与年末资产组合账面持仓完全相同的闭合路径下，传统静态指标判定三人毫无差别，但路径积分的迪维西亚指数却精准揭示出 Manager 2 毫无作为（指数为 1:1）、Manager 1 交易巨亏但依赖外部资金申购掩盖劣质业绩（资产聚集者）、而 Manager 3 创造了超额 Alpha 收益却因赎回缩减规模。完整群在此并非误差，而是评估真实动态特征的核心。Weinstein 借此批评理论物理学界陷入了脱离实际的“修道院文化”，呼吁物理学者打破学术隔绝，将粒子理论、微分几何与经济机制结合为三位一体的“罗塞塔石碑”，重新审视现代几何工具在自然科学与人工系统间的通用法则。 Weinstein 主张市场机制中自然蕴含了等同于基础物理学的规范几何结构，迪维西亚效应和完整群应当作为真实世界的固有特征被接纳。然而，该框架在引入全新商品时仍依赖虚拟影子价格或特征投影等妥协手段，且打破主流经济学对稳定偏好的教条设定仍面临理论范式转换的现实阻力。

[12:42] Eric Weinstein: The mystery here is that natural examples of fiber bundles are ubiquitous but they're ubiquitous within the areas in which we're used to finding them and they're all but undiscovered elsewhere.

[0:00] now um but he remains somebody deeply [0:03] interested in mathematics and [0:04] mathematical physics and he's going to [0:07] be telling us today about the [0:09] relationship between gay Theory physics [0:12] economics and other [0:14] things thank [0:16] you so um and if you want to point [0:26] wel so the topic today um that I'm going [0:30] to try to [0:31] address uh is going to have it as core [0:34] uh gauge theory of [0:37] markets and many of you have probably [0:39] seen at some point in your life um one [0:42] of these talks were somebody from [0:44] psychology department or an operations [0:46] research Department attempts to convince [0:48] you that something U that you know well [0:51] that is very beautiful has tremendous [0:54] application and very [0:56] often you're left feeling that either [0:59] the example is [1:00] tortured um that somebody's trying to [1:03] fit a round peg into an oval [1:05] hole and what I'm hoping today is [1:09] that you will apply a higher standard to [1:12] what it is that I'm going to show you I [1:14] would like to show you an application of [1:16] gauge theory on par to the naturality of [1:20] gauge Theory fiber bundles and [1:21] Connections in uh mathematics physics [1:27] so the point for physics was will be [1:30] what is the implication if a system [1:32] which is not god-given like a [1:35] market lends itself to a description by [1:39] such a structure that we have commonly [1:41] associated with perfect or near perfect [1:45] systems so the inflation will not be uh [1:48] a cosmological theory but will be [1:50] inflation that we're familiar with from [1:52] the [1:54] markets so let's begin with where I [1:57] think a lot of this mysticism started [2:00] which is a beautiful paper by Eugene [2:03] wigner in which probably the most [2:05] beautiful part of the paper is the [2:07] title um why is it that pi and E and Gan [2:13] distributions find themselves uh far [2:16] outside of the Realms in which they were [2:17] originally developed to solve problems [2:20] so wigner said um and I quote here that [2:24] the miracle of the appropriateness of [2:25] the language of mathematics for [2:26] formulation laws of physics is a [2:28] wonderful gift which we neither [2:29] understand understand nor [2:31] deserve um it's a beautiful sentiment I [2:33] may take issue with [2:35] it uh but he clearly saw that it was [2:38] possible that this unreasonable [2:39] Effectiveness might lend itself far [2:42] outside uh of fundamental physics even [2:46] though the unification of gravity and [2:49] the other forces was U really in his [2:52] mind in that paper around [2:56] 1975 something interesting happened at [2:58] ston Brook [3:01] um this was the discovery of the wuyang [3:04] dictionary now I tried to do a little [3:06] bit of research on this and I was [3:08] stunned that there's very little that is [3:10] googleable somehow this topic uh has [3:14] faded from memory so this is the wuyang [3:17] dictionary and memory [3:20] serves James Simons of the math [3:22] department was invited to give lunch [3:24] seminars to some of the physicists and [3:27] something weird happened in the [3:28] cafeteria down there because what they [3:30] found was that there was a complete [3:34] correspondence between Concepts that had [3:37] originated in one field with Concepts [3:39] that had originated in another and it [3:42] was quite striking and mysteriously [3:45] shortly after this um James Simons [3:48] disappears from mathematics and shows up [3:51] as a billionaire on Long Island so it is [3:54] possible that some of the discoveries [3:55] that as much as I would like to [3:57] associate them with myself and my [4:00] collaborator and wife Kia [4:03] Milani it may be that somebody else has [4:05] seen this [4:08] before and more [4:14] successfully so what did Wu and yang add [4:16] to this mix well they said it is a [4:20] widely held view among mathematicians [4:22] that the fiber bundle is a natural [4:23] geometric concept since gauge Fields [4:25] including particular electromagnetic [4:27] field or Fiber bundles all gauge fields [4:30] are thus based on Geometry in the [4:32] underling sentence which followed Begins [4:35] the kind of thinking that we have all [4:36] either accepted or resisted which is [4:39] that some sort of a miracle is going [4:42] on in that the perfect sense the sense [4:46] of perfect modeling where what we are [4:48] trying to model in physics is exactly [4:50] captured by some structure introduced [4:52] for its own Beauty and Mathematics [5:00] this suggests that if we look at these [5:03] curvature equations either linear in the [5:06] curvature or taking a derivative of the [5:09] carer that somehow we are capturing a [5:11] tremendous amount of the natural [5:14] world by simply finding equations that [5:17] we see nowhere [5:19] else and using them to model all of the [5:22] interactions that we believe to govern [5:24] our world is there anything comparable [5:28] in economics well no things are terrible [5:31] in economics from many perspectives and [5:34] wonderful from many others but let's [5:36] look at the case of [5:38] hyperinflation right this is something [5:40] that's had a huge impact on our world [5:41] think about Germany during the early 20s [5:43] and what happened afterwards it's [5:45] probably worthwhile developing a very [5:47] good tool to measure [5:51] inflation the index number [5:54] problem has to do with the fact that [5:57] many economists have introduced me meth [5:59] methods of measuring inflation which are [6:02] not [6:03] compatible so the names most typically [6:06] Associated are L and [6:08] Posh who got in on the ground floor L [6:11] paars index let's see if this is a price [6:15] index what it [6:17] does so let me [6:22] just in what follows we are going to [6:25] consider goods and services [6:35] as [6:38] model by an n-dimensional Vector space V [6:41] over the [6:43] real so implicitly that means that we [6:46] are going to have units for each good [6:50] and we are going to be able to get down [6:52] to fractional amounts of those goods to [6:53] the point where it Continuum Mak sense [6:55] for modeling the [6:58] goods so [7:00] will always be a basket of goods and [7:02] services and in some cases we'll want to [7:04] take Q to have negative components uh [7:07] for example when we're looking at [7:08] currency trading if I go long one [7:10] currency I go short another currency I [7:12] gain one currency by taking on debt [7:14] somewhere else in many cases we'll be [7:17] simply looking at the positive orthant [7:19] or the positive quadrant whatever we [7:21] want to call it of the vector [7:23] space [7:25] conversely if I have a unit such as the [7:29] dollar [7:33] Canadian [7:35] and I attempt to evaluate the price of [7:38] that basket at any given time what I [7:41] have is a [7:43] covector although maybe the economist [7:45] don't frequently say it that way the [7:52] value of p on [7:55] Q is simply the inner product [8:01] given from the vector space dual Vector [8:03] space [8:06] relationship and that's valued in your [8:08] unit of [8:11] currency so what these are our Vector [8:13] relationships in [8:16] components and the Posh and the L fairs [8:21] are the comparison of two vectors two [8:23] co- vectors of prices using either a [8:26] reference Vector from the original time [8:29] in the case of the Les pairs index or [8:31] from the reference or current time in [8:33] the case of the Posh now currently or up [8:36] until recently the lirs index was used [8:40] for the CPI of the United [8:43] States what is the problem with that [8:45] well one what would happen if you used [8:48] any other number key issue is is that [8:52] tax tax brackets are indexed to the cost [8:55] of [8:57] living and in entitlements are indexed [9:00] to the cost of living so if there is [9:03] even a hair's breath of difference [9:04] between one index number and the other [9:06] it creates giant transfers of [9:10] wealth what happened in 1930 was that a [9:13] fellow um to get the first Swedish Bank [9:15] Memorial prize whatever they call the [9:18] that thing um regnar frish proved that [9:23] if you took all of the axum which Irving [9:26] Fisher developed in the 1920s as [9:28] reasonable axum for an index number to [9:30] follow that there was no index number [9:33] that could be constructed which [9:35] satisfied even three of these basic AES [9:38] so not only do you have disagreement but [9:40] you have a theorem that says agreement [9:42] and res resolution of the index number [9:44] problem is [9:48] impossible now as an [9:51] example we can note that if the initial [9:55] price Vector equals the final price [9:57] Vector the top four indic [10:00] yield Unity whereas the divisia index [10:03] does not necessarily yield Unity that is [10:06] an example of a discrepancy so let's [10:08] look at it slightly more closely the [10:10] divisia solves the transitivity problem [10:13] whereas the last pairs solves the [10:14] cycling problem what's the cycling [10:17] problem you have some price which is [10:20] relatively low for some good you're [10:22] consuming uh only a small quantity of a [10:25] Time zero and because the price is low [10:27] and stable you start building your life [10:29] around around it more and more you [10:31] develop a taste for Bordeaux wine [10:32] because it's relatively cheap suddenly [10:35] the Japanese discover bordo wine and the [10:38] price [10:39] climbed it takes you a while to figure [10:41] out that you're wasting money so you [10:44] start switching from Bordeaux over to [10:48] California [10:50] wine your consumption declines and then [10:53] the price comes down as the Japanese [10:54] economy tanks well what's [10:57] happened what's happened is that the L [11:00] pairs index sees no good at the [11:05] beginning the same price at the end and [11:08] it shows no difference what does the [11:11] divisia index [11:13] see during this period of time it [11:15] records no change in the price during [11:18] this period of time it sees you as being [11:20] hit with a price [11:21] increase it sees no change in the price [11:25] and then the price comes down but you [11:27] have no exposure [11:30] so the divisia index doesn't yield Unity [11:32] this is called the cycling [11:35] problem on the other hand if I stop it [11:38] right in the middle and I ask what [11:40] happens if I multiply the less pairs [11:43] index between time one time zero and [11:46] time a half and time a half and time [11:49] one what happens is that that's not comp [11:52] uh that's not compatible with going the [11:54] whole distance from 0 to one whereas in [11:56] the case of the divisia index it behaves [11:58] well under [11:59] composition so you see this is the kind [12:02] of conflict that F was trying to to look [12:06] at how do we try to see where there [12:08] might be opportunity for [12:11] us so we start looking back at calculus [12:14] and the most basic form of calculus [12:16] calculus 1.0 if we think about it in [12:18] terms of releases you're in the XY plane [12:22] and Mathematics is only reasonably [12:24] effective because we've become quite [12:25] jaded about applications and Calculus if [12:27] I tell you that I have a way of [12:29] maximizing the volume of water in a pool [12:31] subject to certain constraints and I'm [12:33] using calculus you're not going to [12:35] object the calculus was developed for [12:37] Celestial mechanics and shouldn't be [12:38] used to figure out an Optimum pool in [12:40] someone's backyard we're all used to [12:42] this idea why because the x- axis and [12:46] the Y AIS in standard calculus are [12:49] measured in real numbers we're all [12:50] comfortable with real numbers we have a [12:52] guaranteed notion of [12:53] horizontal and the idea of taking [12:56] tangents by now seems quite natural [13:01] now it is odd that mathematics which [13:04] finds complex numbers natural and [13:06] physics which finds complex numbers [13:07] natural are still relatively [13:09] distinguished in that most Fields do not [13:12] force people to see a single complex [13:14] number at any time so the reals are very [13:17] important to the story so let's look at [13:19] calculus from the point of view of a [13:22] software company where we're going to [13:23] issue [13:25] upgrades so we have the original [13:27] differential calculus and we have [13:29] multivariate calculus which I'm going to [13:31] call [13:33] 1.1 we then have the absolute [13:36] differential calculus of moving frames [13:38] and we pick up a major client uh with [13:41] work of Einstein and Grossman company um [13:44] in general relativity and we go to [13:46] covariant differentiation in which we [13:49] have internal quantum numbers internal [13:52] symmetry groups and yang n Theory which [13:54] extends enm to the other remaining [13:57] forces and finally we have some sort of [13:59] gauge field theory in which we have [14:01] function spaces and we are no longer in [14:02] finite dimensional territory but we are [14:04] frequently in infinite dimensional [14:06] territory with infinite dimensional [14:08] groups now to some [14:10] extent some of these things make us feel [14:14] that mathematics and physics are really [14:16] special for example if I tell you that [14:18] diff S1 has an important and [14:21] distinguished role in [14:23] economics that's a bit of an eye opener [14:25] because many of us think that's the kind [14:27] of a group that you're only forced to [14:28] deal with with in something as crazy as [14:30] strength Theory but hopefully I can [14:32] convince you that it's not entirely [14:34] unnatural so the mystery here is that [14:36] natural examples of fiber bundles are [14:40] ubiquitous but they're ubiquitous within [14:43] the areas in which we're used to finding [14:44] them and they're all but undiscovered [14:47] elsewhere so we have to ask ourselves is [14:49] the implication that mathematics and [14:52] physics have a special [14:55] relationship or is the implication that [14:57] we have simply failed to follow uh our [15:01] forefathers and foremothers in finding [15:05] highlevel applications of what should be [15:08] relatively Universal gadgets so the [15:10] question is do other important examples [15:13] simply not exist and are these Concepts [15:16] like vector bundles or internal symmetry [15:18] groups and quantum numbers emergent or [15:21] are they really truly fundamental in [15:23] basic and so hopefully by trying to get [15:26] a feel for for the answers to these [15:27] questions we might actually shed light [15:29] back on something as fundamental as [15:31] theoretical physics so first of all [15:34] let's think about gauge theory is [15:36] calculus on fiber [15:38] bundles first problem is where am I [15:42] going to find things that look like [15:45] SpaceTime nice manifolds with good [15:48] properties not necessarily [15:50] coordinati that's not so hard we see [15:53] examples of that multivariate calculus [15:55] where am I going to find interesting [15:57] fibers the fibers that we get to deal [15:59] with in physics are truly interesting [16:02] and some of them in mathematics are very [16:03] interesting as well uh even [16:06] mathematically distinguish so that you [16:07] have sporadic and exceptional [16:11] phenomena the other part is that we have [16:13] to figure out what's going to determine [16:15] the horizontal connections aren't [16:17] interesting unless at some level they're [16:21] non-trivial rather they're not [16:23] interesting Beyond standard calculus [16:25] unless they can be non-trivial so we [16:27] have to imagine what it is it's going to [16:29] pick out interesting connections so in [16:32] this talk I'm going to be talking in [16:33] more of a leev Chita context in which [16:35] the truly interesting connections are [16:37] going to be god-given given some choices [16:40] of structure but the next step is to [16:42] find connections that are given by [16:44] equations and set [16:50] dynamically all right so what is the [16:52] plan the plan is is [16:55] that we're going to try to take over all [16:57] of Economics by going to the heart of [16:59] the matter we're going to look at the [17:00] most basic theories and we are going to [17:03] try to see if we can upgrade them now [17:05] what happened years ago is that [17:07] marginalism swept through economics what [17:09] was marginalism it was the introduction [17:11] of the calculus into the analysis of [17:14] Market [17:15] Behavior so if we have an upgrade of the [17:17] calculus there's nothing to suggest that [17:20] we mathematicians and physicists can't [17:22] swarm into economics and rip out every [17:26] place that they've got one of these [17:28] oldfashioned calculus machines and [17:30] replace it with this new thing uh to [17:33] solve problems that they haven't been [17:34] able to solve so that's the plan I think [17:37] we can take these guys that's my that's [17:38] my feeling and I could use your [17:41] help all right so what do we have we [17:44] have to understand how they see the [17:45] world so how do they see the world well [17:48] they've got these Vector spaces very [17:50] often they throw away everything outside [17:53] of the fundamental uh quadrant where all [17:55] posit all entries are positive [17:59] they foliate so there is usually a [18:06] foliation by hypersurfaces [18:15] of co-dimension [18:19] one which are convex to the origin [18:29] that will be [18:30] called an indifference [18:35] map I will denote by [18:37] o an element of the space of all [18:40] indifference Maps there's some niceties [18:42] here they don't like local minim and [18:45] maximum they like this to be ever [18:47] increasing as you go away from the [18:49] origin so they call these things Bliss [18:52] points um so we're going to foliate this [18:55] positive orth by IND different surfaces [18:59] and then we are going to look at the [19:02] inverse [19:04] images given a particular choice of [19:07] price for every value of the currency in [19:11] which we are denoting the [19:13] prices these are called budget [19:15] constraints so the space of budget [19:18] constraints are a Aline hyperplanes of [19:22] codimension [19:23] one translated from the origin which are [19:27] all based on the current of the pricing [19:31] V before you go on [19:36] us so the indifference curve is a [19:40] measure of a preference that some person [19:43] has [19:46] for right which is not due to my [19:50] insensitivity given Le dietary [19:52] restrictions but the bosin commission [19:54] chose chicken and beef as an example of [19:58] why people buy different quantities of [20:01] different things as the [20:04] angle of the budget constraint [20:08] changes what happens is is that people [20:10] buy different quantities over here [20:13] trying to maximize their utility to get [20:15] onto a Higher indifference surface [20:17] expending their budget constraint we're [20:19] going to get to this in a [20:21] second what was interesting is that the [20:24] bosing commission could not identify any [20:25] other reason why people would switch [20:27] away from beef to chicken other than as [20:29] a response to changing prices and that [20:31] becomes an important part of our [20:33] story so what [20:35] leaves after here is what in biology [20:38] they have this great term the central [20:40] dogma as to [20:41] how um DNA codes for protein and there [20:45] is a central dogma in this story which [20:49] is that people buy the unique [20:54] basket whose tangent at the basket to [20:58] the indifference curve whose tangent [21:00] space is equivalent to the kernel of the [21:03] pricing Vector in [21:06] force they wouldn't say it like [21:09] that so the idea is that the inner [21:12] product of the vector with its duel [21:15] expends the budget constraint and they [21:17] optimize by getting these subspaces to [21:22] coincide the space of all such baskets [21:26] given any particular level of EXP [21:30] expend but with one pricing system in [21:32] mind is called the angle [21:36] curve there is no marks [21:39] curve is that the same I I don't think [21:44] so I'm pretty sure they would have [21:45] driven [21:47] around um okay so this leads to the [21:51] problem of the overstatement of the cost [21:53] of living and you want to talk about an [21:55] expensive problem with the [21:59] what happens if you consume at Point a [22:00] under a pricing regime that gives you [22:03] this angle right you got a budget [22:05] constraint going through a suddenly [22:08] prices [22:10] shift if you give me the money that I [22:13] used to previously spend on my basket at [22:16] a am I going to buy a again no what [22:21] basket I'm going to buy is going to be [22:23] basket C because that gets me onto a [22:26] Higher indiff surface I'm expending the [22:27] same amount of money so if you [22:30] compensate me based on repurchasing my [22:32] basket under the less pairs index I'm [22:35] going to make myself much better off and [22:38] what you should have done is compensated [22:40] me relative to basket [22:41] B because I get a savings from being [22:44] able to substitute so you only have to [22:47] adjust me here in order to keep me at my [22:50] same level of quality of [22:52] life just ask these curves are equal [22:56] amounts of happiness [22:58] more happiness is that way it sounds [23:00] dorky but it's exactly the way they [23:03] think about it so the idea is is that [23:05] they talk about Bliss and happiness and [23:07] utility and they used to talk about it [23:09] in terms of oodles right if you remember [23:11] John Stewart Mill so the idea is that [23:13] once upon a time we were imagined as to [23:16] be giving off you know you could measure [23:17] by the angle of the corners of my mouth [23:19] how happy I am and you write those [23:21] things down and you try to keep me at [23:23] that level of utility but a guy named [23:25] paredo decided you know what this is [23:27] very Sil we can't measure this kind of [23:30] psychic happiness we don't have [23:31] functional MRI yet back in the early [23:33] part of the century so what we're going [23:35] to do is we're going to throw away the [23:37] Cardinal labels and we're going to be [23:39] left with these ordinal [23:41] utility functions in other words we're [23:43] forgetting the cardinality and just [23:45] keeping the foliation so as Lee points [23:48] out this is a constant level of [23:51] happiness or a level of indifference [23:53] hence the name indifferent [23:56] surface but the distinction between [23:58] being at a or being at B in the same [24:00] curve means I could have spent so much [24:03] chicken less beef or more beef less [24:05] chicken I'm just as happy I'm just as [24:07] happy anywhere on that curve but if I [24:09] get more money I'm going to jump up to [24:11] where curve SE is because I want more [24:14] stuff you want to make yourself it's not [24:17] necessarily even more stuff it's that [24:20] combination gives me more [24:24] happiness no I mean because you have [24:25] some sometimes you have some you have [24:27] left very funny effects that I don't [24:29] want to get having more money means I [24:31] can go to a higher per further so you [24:34] have there's a rule that says it has to [24:36] be increasing with respect to the origin [24:39] so that more money never makes you [24:40] miserable whether or not that's true or [24:41] the real world I leave to [24:44] you so this cost of living [24:48] overstatement is what concerns many [24:51] people because of the sheer magnitude of [24:53] it what do I mean by [24:55] that it's a good paragraph to read [24:58] because if we can solve this and we can [25:00] have any impact it's probably worth [25:02] watching the upward CPI bias by itself [25:05] would constitute the fourth largest and [25:07] this is US Federal Outlet program behind [25:09] only social security health care and [25:11] defense that is the CPI [25:14] problem would be the fourth largest [25:17] program if such a thing were considered [25:19] a [25:20] program so any improvements should be [25:24] welcome just just to restate so so the [25:26] problem is that real people will go [25:30] along the present difference curve and [25:32] save money but the Consumer Price Index [25:34] is computed under the assumption that [25:38] they'll they won't do that and they'll [25:40] therefore increase their happiness at [25:41] the same budget it's disgusting isn't it [25:44] but that's exactly right they are going [25:46] to go off and take the money that we [25:48] paid [25:49] them to P to buy their old basket and [25:52] when as soon as we turn their back our [25:53] backs they're going to go and buy a [25:55] completely different basket making [25:57] themselves better off [25:59] also the indexing of the tax brackets [26:02] enters into the picture so those are the [26:03] two effects that really get you into big [26:06] money so it's the fact that the economy [26:08] is hooked up to this number in such an [26:10] intimate fashion that really causes the [26:13] expenses to add [26:18] up so now we have a problem if we're [26:20] going to use indifference curves rather [26:22] than just this mechanical index number [26:25] that we were looking at before like the [26:26] less pairs we have to ask is it [26:29] realistic now what happens if we try to [26:34] incorporate changing [26:37] tastes right what if my tastes don't [26:39] remain stable over a fouryear period so [26:42] you know the famous question are you [26:43] better off now than you were four years [26:44] ago try asking an economist that [26:46] question with the idea that tastes may [26:48] change during the period in question [26:50] what's the problem well if at time one [26:55] point a and point B leave you [26:57] indifferent [26:58] and at time too your tastes have changed [27:00] so that this new [27:02] curve um intersects B and C then B and C [27:06] leave you indifferent at time one but [27:09] point a is clearly less than Point C [27:12] because they're collinear so you start [27:14] to get into all sorts of problems of [27:16] logic as you allow these curves and [27:18] surfaces to cross now what is the [27:21] defense mechanism of a field that knows [27:24] where its vulnerabilities [27:26] are well look at this definition from [27:29] Becker uh Gary Becker who I've never met [27:33] but I do think about him a lot uh is [27:36] very famous for trying to make sure that [27:39] no one ever raises the issue of changing [27:42] tastes this is why I need your help [27:44] because Gary likes to collect shiny [27:46] metals and it's very hard to unseat The [27:50] Chicago School but it what I'm hoping to [27:53] convince you of is that this school is [27:55] so silly with respect to this as to rent [27:58] the theory really [28:00] unusable so he says the combined [28:02] assumptions of maximizing Behavior [28:04] market equilibrium and stable [28:07] preferences so he's encoding stable [28:09] preferences which don't shift into the [28:10] very definition of economics uh in an [28:13] attempt to spike out this [28:16] problem so the most famous paragraph on [28:19] this [28:20] uh I'll let you read it so you don't [28:22] think I'm making it up and I won't [28:23] editorialize [28:30] it's really that that [28:34] way mcf's point of view is more [28:37] sympathetic but very much um the rarer [28:39] point the problem with changes of taste [28:41] have remained Central and largely [28:42] unresolved for 2500 years no doubt [28:45] making some economists think it is wise [28:47] to Define them out of the discipline at [28:48] whatever cost in realism and relevance [28:51] now the real problem which I'm hoping to [28:53] convince you of is that they don't have [28:55] the right mathematics to look at the [28:57] problems and that what they fear is that [28:59] this is going to bounce them out of [29:01] relevance in making policy prescriptions [29:04] but what they really don't understand is [29:05] that they've done a pretty good job and [29:07] that they're fearing that someone is [29:09] going to introduce this problem without [29:11] introducing the appropriate tool so we [29:13] have to ask what would a solution [29:16] mean so you have preferences at period [29:19] one and you have preferences at period [29:21] two and again we're going to take the [29:23] bosin commission's idea that the only [29:26] reason that you would change your [29:27] chicken and beef consumption couldn't [29:29] possibly because you're you're now [29:31] health conscious and you're cutting down [29:32] on red meat all it can be is that it's [29:35] coming from a price change well we're [29:37] going to say no that could possibly be [29:39] coming because of Shifting tastes and so [29:41] we're going to allow any two maps that [29:43] fit this description and what does it [29:45] mean to solve the changing preference [29:47] problem it means to find a mapping of [29:50] the set indexing the foliation at Time [29:53] Zero to the set indexing the foliation [29:55] at time one without essenti reference [29:58] toles if you allow me to label these [30:00] things by numbers then I ask what's [30:03] wrong with saying hey what curve B makes [30:06] me seven Noles happy and here curve D [30:09] makes me seven Noles happy so I'm going [30:11] to call those curves to be in [30:14] correspondence using the [30:17] labels so this requires an intrinsic way [30:20] to match curves on the left those on the [30:21] right without using numerical Cardinal [30:24] labels [30:26] Okay so [30:28] let [30:32] me this is the Eraser there is an eraser [30:36] right [30:40] there but we [30:43] also [30:45] okay [30:47] perfect [30:50] okay so what I'm going to try to do is [30:52] to show you that we can solve the index [30:54] number problem and the changing [30:56] preference problem and the CPI [30:58] uh overstatement problem using guge [31:01] Theory so that's the game in order to do [31:03] that I'm going to introduce three Vector [31:07] bundles no I'm going to introduce three [31:09] bundles two of them Vector one of them [31:11] principal and I'm going to try to show [31:13] you why I believe that this is the right [31:15] way to [31:17] go so let's begin with the less fairs [31:20] Index P not [31:24] sorry P1 [31:28] Q KN dividing divided [31:33] p. [31:35] q1 do not there we go all right so I'm [31:40] looking at prices at time [31:42] one over prices of Time Zero and I'm [31:46] using the reference Vector from time not [31:49] uh to do to uh to weigh both of them now [31:53] I should point out that to every price [31:55] index there is a sort of dual quantity [31:57] index in this case uh it's going to be P [32:03] KN q1 over p [32:07] q so this is a price [32:10] index and this is a quantity index both [32:14] of them lost [32:19] pairs if I did the Posh I'd be using [32:23] current time weights Okay now what's [32:26] wrong with this to a gauge theorist [32:28] what's wrong with those two a physicist [32:30] the key problem here is what is this [32:34] thing here doing at time not being [32:36] multiplied by something at time one how [32:39] did it get from one place to another [32:41] well we parallel transported it using [32:43] some kind of a trivial [32:46] connection why use a trivial connection [32:49] does it make sense we have to find the [32:51] economic connection so here [32:56] goes so what I'm I'm going to try to do [32:58] is I'm going to try to give [33:02] you [33:05] Theory [33:07] one it's going to be based on a base [33:10] space which is going to be V cross V [33:15] Star minus some kind of a singular [33:20] set namely those combinations in which v [33:24] a vector in V and a vector in V Star [33:26] annihilate each other right what is my [33:29] fiber going to be my fiber is going to [33:33] be V cross V Star my total space the [33:38] trivial [33:49] bundle okay what connection do I start [33:55] with first connection I start with is [33:57] the trip [34:01] connection but what happens to this [34:03] bundle F1 sitting over this space it [34:07] admits a decomposition into sub [34:09] buttons [34:11] right so the idea is I've got in inside [34:14] of V I've got my Vector q and I've got [34:18] my n minus one dimensional Subspace per [34:23] P so I've decomposed V into a one [34:27] dimension dial and N minus one [34:28] dimensional [34:30] Subspace same thing is going to happen [34:32] in V Star economists have names for [34:35] these things they don't think of them in [34:36] the same terms we do but they call this [34:39] the income [34:43] effect and they call this space the [34:47] substitution [34:51] effect so far I haven't added anything [34:54] that they don't give you already you're [34:55] always looking at baskets and you're [34:56] always looking at prices [34:59] so what's the real connection that I use [35:02] well the real connection that I use is [35:05] going to be the sum of the [35:09] projection so if the [35:11] fiber [35:14] decomposes as the [35:17] sum of [35:20] subspaces [35:22] um let me call it e for [35:25] effect I I equals [35:28] 1 to [35:29] 4 if the fiber decomposes then I pre and [35:32] post multiply by a projection MTH that [35:35] I'm [35:39] given the trivial [35:45] connection so I sum that [35:49] up now income effects are not mixed with [35:52] income effects nor substitution effects [35:54] with substitution effects same thing in [35:56] the Dual space V Star [35:58] I compare income with income [35:59] substitution with substitution I have a [36:01] notion of parallel translation I then [36:04] attack these indices and I make sure [36:07] that I never multiply [36:08] something uh across time without [36:13] transporting it properly so if I want to [36:15] use Q as a weight I take the value of [36:18] the vector P1 I use the economic [36:21] connection and I parallel translate it [36:24] back so the idea is is that I use the [36:27] parallel [36:32] translate from Time 1 to Time [36:36] Zero using the economic [36:41] connection what happens to the index [36:44] numbers they all become [36:47] coincident as I go ahead so what's this [36:50] one through four is that an example of [36:53] something or what are these four I've [36:55] got two Vector spaces V and V Star in [36:58] the fiber let's look at V first if [37:01] you'll allow me that the point in the [37:03] base space allows me to decompose into a [37:05] onedimensional and an N minus one [37:07] dimensional complement that's two of the [37:09] four the fact that it occurs in the [37:10] other is the second two of the four and [37:13] then my next question is can you try to [37:15] explain in a more intuitive way what [37:18] this economic [37:20] connection be doing so you're moving [37:22] around in this space basically cross [37:25] star Bas space right that's like changes [37:29] in in pricings and changes in quantities [37:33] and then you want to be able to lift [37:35] that PA up to little space and but can [37:39] you say that lift is like what someone [37:41] would so the idea is I'm going to [37:46] allow the modernization of the vector if [37:49] I'm looking at what I purchased in 1960 [37:51] to compared to what I purchased in 1990 [37:54] I going to modernize the vector in terms [37:56] of the ratio of the components purchased [37:59] but at each point in time I'm going to [38:01] say to preserve the proper notion of [38:03] length let's say I'm going to only allow [38:06] it to change by economically allowed [38:08] substitution effects so I'm only going [38:10] to allow the vector I'm going to tell it [38:12] to modernize properly so that if I was [38:15] buying an [38:15] abacus uh you know in the year zero and [38:18] I'm buying a computer in the year 2000 I [38:21] have to cut down on my advocus purchases [38:24] and move towards computer purchases but [38:25] at each point in time as I change my [38:28] consumption [38:29] pattern I'm going to only do it by a [38:32] substitution effect so I never add value [38:35] at any moment or a lose value at any [38:40] moment I don't know what a substitution [38:44] effect if you're buying a small quantity [38:47] of California wine and a large quantity [38:49] of [38:49] Bordeaux right and the idea is that you [38:52] want to move towards California [38:54] wine but you don't want to change your [38:56] total wine purchase that would be a [38:58] substitution effect the amount that you [39:00] increase your California purchase will [39:02] be the amount that you decrease your [39:03] Bordeaux purchase at current price so [39:06] substituting at any given moment is [39:08] clear enough what this says is at each [39:10] moment you're allowed to substitute to [39:12] modernize the vector but no other [39:13] operation is tolerable and in the [39:16] decomposition of those subspaces you [39:18] wrote little Q is little q q not where [39:21] did I write little Q right there beside [39:23] your left right [39:24] hand right there that little relative to [39:28] Q or [39:29] q1 or it's it's Q it's Q an element of [39:33] V it's whatever basket I'm looking at so [39:36] it's going to go from q0 to q1 yeah it's [39:39] going to be there's going to be a path [39:42] there's going to be a path and that [39:43] Vector starts q0 and ends right there's [39:45] going to be [39:47] some some [39:50] path [39:54] TR there's going to be some path in the [39:56] base [39:59] and there's going to be some Vector [40:02] mirroring it in the FI and it's going to [40:06] be allowed to change by substitution [40:09] only that's a choice you make to [40:12] Define theory one yeah it's not much of [40:15] a choice in that it's it's god-given [40:17] structure and the idea is I don't want [40:19] to confuse income and substitution any [40:21] Economist would say that's a perfectly [40:23] fine thing to do where where where [40:25] they're going to choke is that this [40:27] doesn't have curvature equal zero so [40:29] they're going to have to come to grips [40:30] with their fears of [40:33] homy hom is a great thing we know that [40:36] they don't but they [40:40] will okay so that's Theory one so [40:44] hopefully the idea is that that looks to [40:46] be reasonably natural what does it [40:48] produce it produces the divisia index [40:50] everything converges to the divisia so [40:53] in 1925 without understanding the proper [40:56] context these guys came came up with the [40:59] formula no chance in the next example [41:02] why because we're in the a billion [41:03] finite dimensional [41:07] case you're saying uh less how was that [41:11] less pairs less pairs the less pairs [41:13] index when properly uh made gug clariant [41:17] becomes the DIY yes as does the Posh as [41:20] does the TN [41:22] fist they all [41:26] do so now the point is that you get a [41:29] perfect collapse and you say I wonder [41:31] what this divisia is because before we [41:33] thought it was illegal and that it had [41:35] the worst problem of [41:37] all right okay but you know think about [41:40] flat earth right when somebody tells you [41:43] that it's you know they' found a direct [41:45] rout uh that appears to go over the pole [41:47] it looks pretty silly if you do it on a [41:48] Mercador projection so there's something [41:51] to get used to but once you get used to [41:53] it the hope is that you'll see that it [41:54] is economic it gives you extra [41:56] information that you want [41:58] case [42:04] two finite [42:07] dimensional nonability [42:13] what happens in foreign trade people [42:16] disagree on the value of something [42:18] between two countries often with two [42:20] different [42:21] currencies so now the idea is that we [42:23] have q a and QB for two different [42:26] countries [42:27] so my imports are your exports and vice [42:29] versa in bilateral [42:33] trade and we have two different pricing [42:35] systems PA and [42:43] PB if the currencies don't agree because [42:46] purchasing power parity holds even a [42:48] little bit imperfectly and if our [42:50] Imports and our exports don't agree are [42:52] not valued at zero then what we see is [42:54] that we get complimentary substates [42:57] but this time two dimensional and N [43:00] minus two dimensional so the picture is [43:02] going to go through more or less exactly [43:04] the [43:05] same the base space in this case is [43:08] going to be V cross V cross V St cross V [43:13] St minus the singular [43:18] set the [43:21] fiber is going to be V cross V cross V [43:26] Star [43:28] cross V Star the total space is going to [43:31] be [43:34] trivial the connection is going to [43:36] proceed in exactly the same way you're [43:37] still going to have four subspaces [43:39] you're still going to have projection [43:40] off uh operators and what you're going [43:43] to get is going to be instead of the [43:46] usual divisia index which [43:50] is the [43:52] exponential and I'm going to write it a [43:54] little bit funny here [43:59] PDQ [44:01] PQ inverse because I'm going to put p [44:06] and [44:07] [Music] [44:08] q q [44:10] a [44:14] QB so these are two vectors I'm going to [44:16] put them into a n by2 Matrix and I'm [44:19] going to take PA and I'm going to take [44:23] QA and I'm going to put that into a 2xn [44:26] matrix [44:27] so that PQ is a 2X [44:30] two so I'm going to take the exponent of [44:33] the integral over the path of PQ inverse [44:39] times [44:41] P [44:52] DQ but now I have a [44:55] problem I'm going to have to do [44:56] something some kind of a time ordered [44:58] product to get the exponential otherwise [45:00] I'm reduced to doing things over [45:02] simplies so time ordered product of this [45:05] is going to give me a 2X two [45:08] Matrix which is going to be the Matrix [45:12] value version of the less pairs in the [45:14] Posh index again economics has never [45:16] seen these 2 by two uh valued indices [45:20] but the idea is that we're now in [45:22] g2r and g2r is intrinsically non ailan [45:26] so we get away with the simple exponent [45:28] that they were able to do in the ailan [45:30] case more analogous to electromagnetism [45:33] now we're in something like isotopic [45:36] spin we've got non ailan aspects and [45:38] then we now need a Time ordered [45:42] product the last [45:48] example just a question is this roughly [45:51] the effect that if I have a chain of [45:53] currencies I trade one for one for one [45:56] for one for one you're not assuming that [45:58] you get back to the same place well if [46:01] you're able to do it fast and you can [46:02] get spreads to zero for example by using [46:06] EBS trading system I mean things getting [46:08] really tight so if you can get your [46:10] spreads to zero so that your transaction [46:11] costs are low and you can execute [46:13] milliseconds and you can get what they [46:15] call Choice pricing and then that bid [46:17] and ask are exactly the same you [46:19] shouldn't be able to make money or lose [46:21] money on that [46:23] bet so that is sort of a primitive man's [46:25] notion of homi but we'll see where that [46:29] hold that idea because it's coming up if [46:31] I can just eat the clock here hom is [46:33] when you do make m hom is when you do [46:35] make money homi is a beautiful thing but [46:38] homi can cause you to lose [46:40] money okay last one so this is supposed [46:43] to be a really different example so now [46:46] we're going to do infinite [46:52] dimensional diff S1 [46:58] orientation preserving different [47:01] morphisms of the circle and small change [47:04] I'm going to fix a point so it's really [47:06] dimorphisms with the real line so [47:08] infinite dimensional non [47:12] ailion now we're not going to see a [47:15] vector bundle we're going to see a [47:16] principal bundle come out of [47:19] nowhere what is my base [47:23] space my base space [47:28] I'll put a hat over because I'll change [47:30] it in a second is going to be the space [47:32] of all ordinal preference Maps sbly [47:35] apologize and made continuous [47:39] differential what is my [47:44] group my group is just this [47:53] diff and what is my space the space of [47:56] Cardinal utility functions familiar from [48:00] Mil all right so I have a preference [48:07] map and as long as I decide that there [48:10] cardinal utility functions over it are [48:13] surjective onto the positive [48:16] re then the idea is is that I can come [48:19] up with [48:23] different Cardinal utility functions [48:25] whose contour maps are [48:28] indistinguishable when viewed over the [48:30] space of goods and [48:32] services so these Cardinal utility [48:35] functions these surfaces above the [48:38] ordinal preference MTH as long as they [48:40] have the same Contours I can act on them [48:42] if if the action is s if the functions [48:47] are subjective onto the reels by the [48:49] reparameterizations of the reels that [48:50] are orientation [48:52] preserve when I quo out what do I get I [48:55] get precisely the or preference maps of [48:59] Paro okay what does this bu me well [49:03] let's [49:04] assume that I'm in this crazy infinite [49:07] dimensional function space [49:09] bundle and I've got these groups and I [49:12] look for this [49:15] path and I look for a lift any lift [49:18] doesn't matter that's looking for a Time [49:21] dependent Cardinal utility function C of [49:24] T element [49:28] of the Cardinal utility [49:31] function now what if I have a [49:34] connection if I had a connection I could [49:38] use the distinguished path I could [49:41] project [49:42] down and then I could lift it back up [49:44] using the distinguished path to some [49:48] other lift which would be god-given [49:51] except for the starting [49:54] point right [49:57] so if I can measure how much I have to [49:59] adjust this Cardinal utility function to [50:02] hit the horizontal lift in the [50:05] connection that is going to give me a [50:07] very essential object why because if I [50:10] scale the entire carinal utility [50:12] function it's not going to have any [50:14] impact in other words if I have a [50:16] cardinal utility function at time zero [50:20] right I map I compose [50:24] that with [50:27] I'm not good at drawing oops salons [50:28] forgot what I called it [50:31] recently oops [50:33] salon and I compose [50:36] that with a Time One Cardinal utility [50:39] function and this is based on doing a [50:42] computation to bring the whole picture [50:45] to be the horizontal Li so I have I pick [50:47] any time depend Cardinal utility [50:51] function I adjust it by this [50:55] object and [50:57] I I map into the Oodles I reparameterize [51:01] the uel and I map [51:02] out that operation is insensitive to the [51:06] initial [51:07] choice of cardinal utility function that [51:10] was lifted above the space um of ordinal [51:14] Maps which were actually chosen by our [51:17] our [51:18] consumer so this gadget is a solution to [51:21] the changing preference [51:25] problem all right right and notice no [51:28] these are all objects these two objects [51:31] are the center of their [51:32] theory in this object who can [51:36] object what do we get if we use this [51:40] intertemporal uh welfare comparison is [51:42] we get a consistent notion independent [51:44] of the choice of cardinal utility [51:46] function representing but we don't have [51:48] a connection in fact I conjecture that [51:50] there isn't a natural connection on the [51:52] space that's [51:54] distinguished very sad [51:58] except I have prices so if I cartisian [52:02] product with the space of [52:06] prices this becomes my true space [52:12] B3 [52:14] theorem there [52:16] exists [52:23] a god-given [52:33] connection on this Total [52:42] Space prices are available in the market [52:45] so I'm not adding any new information [52:47] that wasn't available [52:49] otherwise and now I've solved the [52:51] changing preference problem using gauge [52:54] Theory with nothing that these guys [52:57] don't use [52:59] themselves what I ask you as [53:02] businesss aside from your help in [53:04] overthrowing [53:05] Chicago is to try to ask the question [53:09] what does it mean if all of this natural [53:11] structure is just outside our door for [53:13] our own [53:15] theories does it mean I [53:18] mean we know that no deity created these [53:21] markets and yet the fit is as hand and [53:23] glove to my way of thinking as it is in [53:26] the standard [53:27] model so what we're discovering is the [53:30] unreasonable effectiveness of utility as [53:32] a gauge [53:35] Theory we have a fiber [53:38] bundle which is Cardinal utility F uh [53:42] functions cross prices we have honest to [53:44] God curvature we have a solution to this [53:46] problem and this gives us a Rosetta [53:49] Stone namely we have particle theory we [53:52] have economic theory and we have [53:54] geometry [53:56] and I would conjecture that this story [53:58] does not stop here why because what [54:00] we're looking at is the fact that it's [54:02] not that partical physics and geometry [54:06] have this peculiar relationship it's the [54:09] fact that we have started having a [54:11] problem finding applications outside of [54:14] what it is we do and to go a little bit [54:16] further more provocatively I would say [54:18] that we're having a problem with our [54:20] monastic culture integrating the talent [54:24] and the understanding that exists in [54:27] theoretical uh enclaves like this one [54:29] with the real world problems right [54:31] outside our door and this is leading to [54:32] a distorted way of thinking about uh [54:36] theoretical physics and Mathematics as [54:38] well so the key question is what does [54:40] this tell us does this tell us that [54:42] geometry is emergent to connections are [54:45] they fundamental are we looking at [54:47] effective theories are we looking at [54:48] basic theories I think that as we find [54:51] more applications this is going to look [54:53] more commonplace and Lee has asked the [54:55] question in a paper about why are there [54:57] no [54:58] Einsteins well in part the fact is that [55:01] we haven't been applying the tools of [55:03] the trade successfully outside if we [55:05] start to do that the picture may look [55:07] very very [55:08] different so just to give you an idea of [55:10] the formulas what I was calling Upsilon [55:13] here I think is Omega here um I did not [55:17] have the time to get this in form I [55:19] wanted uh before the talk so forgive me [55:23] but the idea is is that it's just going [55:25] to be some sort of Dyson form of a [55:27] perturbation series which gives you this [55:29] adjusted Factor we have this [55:32] construction of the changing preference [55:33] welfare map we have a bilateral index [55:38] which you can compute with no knowledge [55:39] of preferences from the Assumption of [55:41] cob Douglas preferences what is that you [55:43] take good one raise it to the a to the [55:46] alpha power good B you raise it to the [55:49] beta power you raise the last one to the [55:51] one minus the sum so there's a canonical [55:53] change uh preference map when you don't [55:55] know what the answer is and so you can [55:57] interpolate between two observed [55:59] purchases somebody purchased something [56:01] at time one something at Time Zero you [56:03] assume cob dougas presences interpolate [56:05] and you get another index number based [56:07] on the changing preference so this is as [56:09] concrete as you please and lastly we [56:11] have this trade index which is a 2X two [56:13] Matrix valued object coming from a non [56:15] ailon [56:17] Theory so in [56:20] closing I was once told by someone um [56:23] that the Chinese and I don't know why [56:25] the Chinese get credit for all these [56:26] things uh say uh that if you're going to [56:30] try to see the universe in a single [56:32] grain of sand make sure the grain of [56:34] sand is well chosen I think there are a [56:37] lot of things to see here and one of the [56:40] things that would be terrific in [56:41] conversations I'm here till Friday is to [56:45] find out whether there's interest in [56:48] taking the tools that are common to our [56:51] stock and trade here and heading into [56:54] the markets and taking it sort of a [56:56] different point of view as to what [56:59] geometry and physics are really doing [57:01] together I thank you for your [57:02] [Applause] [57:09] time questions [57:13] comments so uh in the change from [57:18] constant preferences to changing [57:20] preferences it appeared to me that the [57:22] number of products available are [57:24] constant in other words can be can be C [57:27] different uh is it possible to [57:29] incorporate if you suddenly tomorrow [57:31] have 15 more choices than you had [57:33] yesterday great question yes it is but [57:37] it's not uh as beautiful as you would [57:40] like so one way to do it is to throw all [57:42] the goods and services that will ever be [57:44] available and in Phantom prices that [57:47] would drive their [57:48] consumption uh at technologically [57:50] unfeasible ages to zero or near zero so [57:54] the idea is yes there's a toaster aail [57:56] able to the Infant Jesus but [57:58] unfortunately the price would be [58:00] prohibited that's one thing you can do [58:02] another thing you can do is you can work [58:03] with honic what iSonic it is the mapping [58:06] again they wouldn't say it this way of [58:07] one vector space to a vector space of [58:09] strictly smaller Dimension with [58:10] non-trivial kernel so the idea is that a [58:13] toaster becomes a food preparation [58:14] device and a skillet at a different [58:16] period of time would be equivalent to a [58:19] toaster through their efficacy in making [58:21] toast uh I leave it to you but there are [58:24] ways of incorporating it um my goal is [58:27] not to solve the problem of how to get a [58:30] toaster to the Infant Jesus it's to try [58:32] to figure out how to expand the [58:34] framework well beyond where it is [58:45] already a quick at the map you [58:51] hads [58:53] colum effect yeah as [59:00] uh [59:01] cycling is it cycling problem [59:05] problem and so that that's I was trying [59:08] to figure out where the quantum verus [59:11] classical the oh you know I didn't show [59:14] that and that's that's a good thought [59:18] so am [59:20] I okay so here's a picture this is Lee's [59:24] question is homi good is homi bad [59:26] why should we accept it and hug it close [59:28] to our so you have a situation here in [59:32] which I'm going to depict [59:34] graphically the one-dimensional fiber [59:36] bundle assume two goods right if I'm if [59:39] I'm going to ction out V cross V Star I [59:41] me I have a four dimensional space but I [59:43] really only need the relative uh [59:46] proportion so I'm going to quo out and [59:48] work with projective spaces in my base [59:50] space so that's going to allow me to get [59:51] into visual territory without loss of [59:53] generality and then I need a [59:55] one-dimensional fiber which is just [59:56] going to be the income effects and [59:58] forget the substitution effects so here [60:01] you are in price quantity space where [60:04] prices let's say are exhibiting simple [60:06] harmonic oscillation [60:08] right we have three hedge fund [60:19] managers so we have manager [60:24] one who theis [60:31] price [60:33] is um two his divisia [60:41] quantity is 1 [60:44] 12 we have manager [60:46] 2 whose divisia [60:49] price is one and his divisia [60:53] quantity is one and we have manager 3 [60:57] whose divisia [61:02] price [61:04] is3 his divisia [61:08] quantity [61:09] is [61:11] three all [61:14] right so they've been in the same market [61:17] and they wound up their position they [61:18] have the same positions that they were [61:20] given and they were told on January 30 [61:22] uh on December 31st get back to the same [61:26] relative components that you started the [61:28] year with we're going to [61:30] evaluate [61:32] now what do you do with these three [61:35] guys first of [61:37] all he gets [61:40] fired so why are you guys going to fire [61:43] him well he sat on his Capital he didn't [61:46] do [61:48] anything and the aggregate his quantity [61:52] index stayed the [61:54] same what are you going to do with [61:57] manager [61:59] one with manager one if he hadn't [62:02] brought in any Capital so he had no [62:04] subscriptions and he hadn't lost any [62:06] Capital because his investors wanted it [62:08] back so he had no redemptions he would [62:10] have gone around this Loop and he would [62:12] have came come back if he'd been [62:14] successful with a higher quantity but he [62:18] came back with the [62:21] same [62:23] composition so if all of these guys [62:27] had P equal to P1 I guess I forgot this [62:31] part if my [62:34] mistake it should be [62:36] little okay so all three of them had [62:39] their initial price vectors and their [62:40] initial quantity vectors [62:42] equal the only way that this guy could [62:45] have come back with a quantity index [62:49] of2 and come back with the same quantity [62:51] that he started was he was successful in [62:53] attracting Capital he's what we call an [62:55] asset Gap in the hedge fund industry but [62:59] he's lost money in trading so what you [63:02] now do is you take manager [63:04] one and you make [63:06] him the sales [63:11] rep for manager 3 who is some kind of a [63:14] missing throat who's very good at [63:15] trading markets but boying people take [63:18] their money away from him as he's making [63:20] the money because he only ended up with [63:22] a the same quantity that he started with [63:25] by virtue of the fact that he went [63:26] around the loop and people took the [63:27] money away in redemptions and said I [63:29] can't deal with that guy so the idea is [63:32] that everybody started with the initial [63:35] prices the same everybody finished with [63:37] the initial prices the same started with [63:38] the initial quantities the same finished [63:40] with the the final quantities the same [63:42] and yet the divisia index gave us true [63:45] information about what happened in the [63:47] system why because the homi is a good [63:50] positive healthy thing that is naturally [63:52] present in the system it is no there's [63:55] no sense in [63:57] flying from London to Los Angeles on a [64:01] direct route via Mercer projection you [64:03] should use the [64:05] pole so at some level the cycling [64:09] problem is no problem at all just as the [64:11] way the B Marino effect is a real [64:13] feature of the world and should be [64:14] embraced rather than [64:21] fear question [64:26] I hope you guys will drop through I [64:28] think I'm in [64:29] 259 if anybody wants to talkk [64:35] [Applause]